Remark 6.8 For the ordinary torus \(n\) , i.e., \(\theta =0\) , by a recent result of Sukochev-Zanin [1]} the estimate (REF )
still holds if \(x\)  is in the Orlicz space \(L\!\log \!L(n)\)  (see also [2]}). It would be interesting to have an analogue of this result for \(\theta \ne 0\) .

